Question: $ C = \left[\begin{array}{rr}5 & 3 \\ 3 & 4 \\ 0 & 4\end{array}\right]$ $ A = \left[\begin{array}{rr}0 & 3 \\ -2 & 5\end{array}\right]$ What is $ C A$ ?
Because $ C$ has dimensions $(3\times2)$ and $ A$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ C A = \left[\begin{array}{rr}{5} & {3} \\ {3} & {4} \\ \color{gray}{0} & \color{gray}{4}\end{array}\right] \left[\begin{array}{rr}{0} & \color{#DF0030}{3} \\ {-2} & \color{#DF0030}{5}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rr}{5}\cdot{0}+{3}\cdot{-2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{0}+{3}\cdot{-2} & ? \\ {3}\cdot{0}+{4}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{0}+{3}\cdot{-2} & {5}\cdot\color{#DF0030}{3}+{3}\cdot\color{#DF0030}{5} \\ {3}\cdot{0}+{4}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{5}\cdot{0}+{3}\cdot{-2} & {5}\cdot\color{#DF0030}{3}+{3}\cdot\color{#DF0030}{5} \\ {3}\cdot{0}+{4}\cdot{-2} & {3}\cdot\color{#DF0030}{3}+{4}\cdot\color{#DF0030}{5} \\ \color{gray}{0}\cdot{0}+\color{gray}{4}\cdot{-2} & \color{gray}{0}\cdot\color{#DF0030}{3}+\color{gray}{4}\cdot\color{#DF0030}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-6 & 30 \\ -8 & 29 \\ -8 & 20\end{array}\right] $